Problem 2 More Than One Solution

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1 Problem More Than One Solution 1. Water becomes non-liquid when it is 3 F or below, or when it is at least 1 F. a. Represent this information on a number line. b. Write a compound inequality to represent the same information. Define your variable.. Luke and Logan play for the same baseball team. They practice at the Lions Park baseball field. Luke lives 3 miles from the field, and Logan lives miles from the field. a. First, plot a point to represent the location of the Lions Park baseball field. b. Next, use your point that represents Lions Park, and draw a circle to represent all the possible places Luke could live. c. Finally, use your point that represents Lions Park, and draw another circle to represent all the possible places Logan could live. 01 Carnegie Learning d. What is the shortest distance, d, that could separate their homes? e. What is the longest distance, d, that could separate their homes?.4 Solving and Graphing Compound Inequalities 115

2 f. Write a compound inequality to represent all the possible distances that could separate their homes. g. Represent the solution on a number line. 3. Jodi bought a new car with a 14-gallon gas tank. Around town she is able to drive 336 miles on one tank of gas. On her first trip traveling on highways, she drove 448 miles on one tank of gas. What is her average miles per gallon around town? What is her average miles per gallon on highways? a. Write a compound inequality that represents how many miles Jodi can drive on a tank of gas. Let m represents the number of miles per gallon of gas. b. Rewrite the compound inequality as two simple inequalities separated by either and or or. c. Solve each simple inequality. d. Go back to the compound inequality you wrote in Question 3, part (a). How can you solve the compound inequality without rewriting it as two simple inequalities? Solve the compound inequality. e. Compare the solution you calculated in Question 3, part (c) with the solution you calculated in Question 3, part (d). What do you notice? 01 Carnegie Learning f. Explain your solution in terms of the problem situation. 116 Chapter Graphs, Equations, and Inequalities

3 g. Represent the solution on a number line. Describe the shaded region in terms of the problem situation. Problem 3 Solving Compound Inequalities Remember, a compound inequality is an inequality that is formed by the union, or, or the intersection, and, of two simple inequalities. The solution of a compound inequality in the form a x b, where a and b are any real numbers, is the part or parts of the solutions that satisfy both of the inequalities. This type of compound inequality is called a conjunction. The solution of a compound inequality in the form x a or x b, where a and b are any real numbers, is the part or parts of the solution that satisfy either inequality. This type of compound inequality is called a disjunction. 1. Classify each solution to all the questions in Problem as either a conjunction or disjunction. Let s consider two examples for representing the solution of a compound inequality on a number line. The compound inequality shown involves and and is a conjunction. x 1 and x 3 Represent each part above the number line. 01 Carnegie Learning x 1 x > 3 x 1 and x > 3 3 < x 1 The solution is the region that satisfies both inequalities. Graphically, the solution is the overlapping, or the intersection, of the separate inequalities..4 Solving and Graphing Compound Inequalities 117

4 The compound inequality shown involves or and is a disjunction. x or x 1 Represent each part above the number line. x < x > 1 x < or x > 1 The solution is the region that satisfies either inequality. Graphically, the solution is the union, or all the regions, of the separate inequalities.. Consider the two worked examples in a different way. a. If the compound inequality in the first worked example was changed to the disjunction, x # 1 or x 3, how would the solution set change? Explain your reasoning. b. If the compound inequality in the second worked example was changed to the conjunction, x, or x 1, how would the solution set change? Explain your reasoning. 3. Represent the solution to each compound inequality on the number line shown. Then, write the final solution that represents the graph. a. x or x 3 01 Carnegie Learning 118 Chapter Graphs, Equations, and Inequalities

5 b. 1 x 1 c. x 0 or x d. x 1 and x Pay attention to whether the inequality uses and or or. e. x 3 and x f. x and x 1 01 Carnegie Learning g. x 1 or x 0.4 Solving and Graphing Compound Inequalities 119

6 To solve a compound inequality written in compact form, isolate the variable between the two inequality signs, and then graph the resulting statement. To solve an inequality involving or, simply solve each inequality separately, keeping the word or between them, and then graph the resulting statements. 4. Solve and graph each compound inequality showing the steps you performed. Then, write the final solution that represents the graph. a. 6 x 6 9 b. x 6 c. 4 3x 1 1 d. x 7 10 or x Carnegie Learning 10 Chapter Graphs, Equations, and Inequalities

7 e. 1 x 3 4 or x 3 f. 1 6x 11 or x Carnegie Learning Be prepared to share your solutions and methods..4 Solving and Graphing Compound Inequalities 11